AP Statistics prep

Covers CED Topic 2.3: random processes, outcomes and events, designing and running simulations, probability as long-run relative frequency, and the law of large numbers.

• Identify the random process, outcomes and events in a chance situation.
• Design a simulation that assigns outcomes to values determined by chance, and record the counts of simulated outcomes and the total number of trials.
• Estimate a probability as the relative frequency of an event in simulated or empirical data.
• Explain the law of large numbers: as the number of independent trials increases, the relative frequency of an event gets closer to a single value.

Covers CED Topics 2.4–2.5: sample spaces, equally likely outcomes, the complement rule, joint probability and mutually exclusive (disjoint) events.

• Identify the sample space of a random process and calculate probabilities when outcomes are equally likely.
• Calculate the probability of the complement of an event, $P(E^C)=1-P(E)$.
• Calculate a joint probability $P(A \cap B)$ from a table or description.
• Justify whether two events are mutually exclusive using $P(A \cap B)=0$.

Covers CED Topic 2.6: calculating $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$ from two-way tables, tree diagrams and descriptions, and the general multiplication rule $P(A \cap B)=P(A)\cdot P(B \mid A)$.

• Calculate conditional probabilities from two-way tables and descriptions using $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$.
• Apply the general multiplication rule $P(A \cap B)=P(A)\cdot P(B \mid A)$ to multistage chance processes.
• Organize sequential events in a tree diagram and calculate probabilities from it.
• Interpret a conditional probability in context, naming the condition.

Covers CED Topic 2.7: checking and using independence ($P(A \mid B)=P(A)$ and $P(A \cap B)=P(A)\cdot P(B)$) and the addition rule for the union of two events.

• Determine whether two events are independent by comparing $P(A \mid B)$ with $P(A)$, or $P(A \cap B)$ with $P(A)\cdot P(B)$.
• Calculate probabilities for independent events using $P(A \cap B)=P(A)\cdot P(B)$.
• Calculate the probability of a union using $P(A \cup B)=P(A)+P(B)-P(A \cap B)$.
• Distinguish independent events from mutually exclusive events.

Covers CED Topic 2.8: random variables, probability distributions of discrete random variables built from probability rules or simulation and shown as tables, graphs or functions, and cumulative probability distributions.

• Define a random variable as a variable whose numerical values result from a random phenomenon.
• Construct a probability distribution for a discrete random variable as a table, graph or function, with probabilities that sum to 1.
• Determine or estimate a discrete probability distribution using the rules of probability or a simulation.
• Construct and use a cumulative probability distribution to find $P(X \le x)$.

Covers CED Topic 2.9: parameters of a probability distribution, the expected value $\mu_X=\sum x_i \cdot P(x_i)$, the standard deviation $\sigma_X=\sqrt{\sum (x_i-\mu_X)^2 \cdot P(x_i)}$ and variance, and their interpretation in context.

• Calculate the expected value (mean) of a discrete random variable, $\mu_X=E(X)=\sum x_i \cdot P(x_i)$.
• Calculate the standard deviation $\sigma_X$ and variance $\sigma_X^2$ of a discrete random variable.
• Interpret the expected value as the long-run average outcome, in context and with units.
• Interpret the standard deviation as the typical deviation of the values of the random variable from its mean over the long run.

Covers CED Topic 2.10: the conditions for a binomial random variable, its mean $np$ and standard deviation $\sqrt{np(1-p)}$, probabilities from $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$ or technology, estimates from simulation, and interpretation in context.

• Justify whether a random variable is binomial: it counts successes in a fixed number $n$ of independent trials, each with two outcomes and the same probability of success $p$.
• Calculate the mean $\mu_X=np$ and standard deviation $\sigma_X=\sqrt{np(1-p)}$ of a binomial random variable.
• Calculate exact and cumulative binomial probabilities with the binomial probability function or technology, naming the distribution, its parameters and the boundary.
• Estimate probabilities of a binomial random variable using data from a simulation.
• Interpret the mean, standard deviation and probabilities of a binomial distribution in context.

Covers CED Topic 2.11 (learning objectives A–C): continuous random variables, the shape and parameters $\mu$ and $\sigma$ of a normal distribution, the standard normal distribution, and the 68–95–99.7 (empirical) rule.

• Describe a normal distribution as continuous, unimodal, bell-shaped and symmetric, identified by its mean $\mu$ and standard deviation $\sigma$.
• Explain how a smaller or larger standard deviation makes a normal curve taller and more concentrated or shorter and more spread out.
• Identify the mean and standard deviation of a normal distribution, including the standard normal distribution with $\mu=0$ and $\sigma=1$.
• Calculate approximate percentages with the empirical rule: about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.

Covers CED Topic 2.11 (learning objectives D–F): areas under a normal curve as probabilities, finding the values that bound a given area (lowest, highest, middle or most extreme $p\%$) with technology or Table A, and comparing relative positions with percentiles.

• Calculate the probability that a normally distributed variable falls in an interval as the area under the normal curve, using technology or z-scores and Table A.
• Calculate the boundaries $x_a$ and $x_b$ for a given lower, upper, middle or two-tailed area of a normal distribution.
• Show a distribution calculation by naming the distribution and its parameters, labeling the boundary values and direction, and giving the answer.
• Compare relative positions of values within or between normal distributions using percentiles and proportions.

Covers CED Topic 2.12: the sampling distribution of a statistic, simulating sampling distributions from a population with known parameters, randomization distributions from reallocating responses to treatment groups, and the central limit theorem.

• Describe the sampling distribution of a statistic as the distribution of its values over all possible samples of a given size from a population.
• Describe how to simulate a sampling distribution by repeatedly sampling from a population with known parameters, and interpret the simulated distribution.
• Describe a randomization distribution generated by repeatedly reallocating response values to treatment groups.
• Explain the central limit theorem: the sampling distribution of the mean of a random sample is approximately normal, and the approximation improves as the sample size increases.
• Distinguish the distribution of a population, the distribution of the data in one sample, and the sampling distribution of a statistic.

Covers CED Topic 3.3: the one-sample z-interval for a population proportion: defining the parameter, verifying conditions (observed successes and failures at least 10), critical values $z^*$, the interval $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, the standard error, the margin of error and the minimum sample size.

• Identify the one-sample z-interval for a population proportion and define the parameter in context (the proportion, the response variable and the population).
• Justify the randomization, 10% and normality conditions, using the observed numbers of successes $n\hat{p}$ and failures $n(1-\hat{p})$.
• Calculate the interval $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ with the correct critical value $z^*$ for the confidence level.
• Calculate the standard error and the margin of error of $\hat{p}$.
• Calculate the minimum sample size for a given margin of error, using $\hat{p}=0.5$ when no estimate is available.

Covers CED Topic 3.4: interpreting a confidence interval and a confidence level in context, using an interval of plausible values to justify a claim, and how the confidence level and sample size affect the critical value, margin of error and width.

• Interpret a C% confidence interval for a population proportion in context, referring to the parameter and the population.
• Interpret the confidence level as the long-run proportion of intervals that capture the parameter in repeated random sampling with the same sample size.
• Justify a claim about a population proportion based on whether the claimed value lies within the interval of plausible values.
• Identify how increasing the confidence level increases the critical value, margin of error and width, and how increasing the sample size decreases the width, approximately in proportion to $\frac{1}{\sqrt{n}}$.

Covers CED Topic 3.5: the one-sample z-test for a population proportion: defining the parameter, writing null and alternative hypotheses (one-sided or two-sided), and verifying conditions with the expected counts $np_0$ and $n(1-p_0)$.

• Identify the one-sample z-test for a population proportion and define the parameter in context.
• Identify the null hypothesis $H_0: p=p_0$ and a one-sided ($p<p_0$ or $p>p_0$) or two-sided ($p \ne p_0$) alternative from the investigative question.
• Explain that the null hypothesis is the status quo, assumed true unless there is convincing statistical evidence otherwise, and that the alternative states the researcher’s claim.
• Justify the randomization, 10% and normality conditions, using the expected counts $np_0 \ge 10$ and $n(1-p_0) \ge 10$.

Covers CED Topics 3.6–3.7: the null distribution, calculating and interpreting p-values from the standard normal distribution or from a simulated null distribution, the test statistic $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, significance levels, the formal decision and a conclusion in context.

• Calculate the test statistic $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ and its p-value from the standard normal distribution, matching the tail or tails to the alternative hypothesis.
• Calculate a p-value from a simulated null distribution as the proportion of simulated values as extreme as or more extreme than the observed statistic.
• Interpret a p-value in context as a probability computed by assuming the null hypothesis is true.
• Make a formal decision by comparing the p-value with the significance level $\alpha$: reject $H_0$ when the p-value is at most $\alpha$, and otherwise fail to reject it.
• Justify a conclusion in context in terms of the alternative hypothesis, using nondefinitive language and never accepting or proving $H_0$.

Covers CED Topic 3.8: identifying, interpreting and finding the probabilities of Type I and Type II errors, the power of a test, the factors that increase power, and how the consequences of each error guide the choice of significance level and sample size.

• Identify and describe Type I and Type II errors in context for a given test.
• Calculate the probability of a Type I error as $\alpha$ and of a Type II error as $1-\text{power}$.
• Identify how a larger sample size, a smaller standard error, a true parameter value farther from the null value or a larger $\alpha$ increases power and decreases the probability of a Type II error.
• Interpret the consequences of Type I and Type II errors in context, and explain how they inform the choice of significance level and sample size.

Covers CED Topic 3.9: the mean $p_1-p_2$ and standard deviation $\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$ of $\hat{p}_1-\hat{p}_2$, the conditions for independent random samples or randomized experiments, and probabilities interpreted for two populations.

• Calculate the mean and standard deviation of the sampling distribution of $\hat{p}_1-\hat{p}_2$.
• Justify the randomization condition (two independent random samples, or random assignment in an experiment) and, when sampling without replacement, the 10% condition for each sample.
• Justify approximate normality using the expected numbers of successes and failures in both samples, all at least 10.
• Calculate and interpret probabilities involving $\hat{p}_1-\hat{p}_2$ in the context of the two populations.

Covers CED Topics 3.10–3.11: the two-sample z-interval for $p_1-p_2$: defining the parameters, verifying conditions, calculating the standard error, margin of error and interval, interpreting it in context, and using whether it contains 0 to justify a claim.

• Identify the two-sample z-interval for a difference between population proportions and define the parameters in context, including the order of subtraction.
• Justify the randomization, 10% (not needed for a randomized experiment) and normality conditions, using the observed successes and failures in both samples.
• Calculate $(\hat{p}_1-\hat{p}_2) \pm z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$, its standard error and its margin of error.
• Interpret the interval and the confidence level in context for the difference between two population proportions.
• Justify a claim about a difference in population proportions, using whether the interval contains 0 or another claimed value.

Covers CED Topics 3.12–3.13: the two-sample z-test for $p_1-p_2$: parameters and hypotheses, conditions using the combined (pooled) proportion $\hat{p}_c$, the test statistic and p-value, and the conclusion in context. The CED’s sample inference question uses this procedure.

• Identify the two-sample z-test for a difference between population proportions and define the parameters in context.
• Identify the null hypothesis $H_0: p_1=p_2$ (or $p_1-p_2=0$) and a one-sided or two-sided alternative.
• Justify the randomization, 10% and normality conditions, using the combined proportion $\hat{p}_c=\frac{n_1\hat{p}_1+n_2\hat{p}_2}{n_1+n_2}$ to check the expected successes and failures.
• Calculate $z=\frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$ and its p-value, and interpret the p-value assuming the population proportions are equal.
• Justify a conclusion about the populations in context by comparing the p-value with $\alpha$, using nondefinitive language in terms of the alternative.

Covers CED Topic 3.14: chi-square statistics and distributions, choosing between the chi-square test for homogeneity (several populations or treatments) and the chi-square test for independence (two variables in one population), writing hypotheses in words, and verifying conditions.

• Describe chi-square distributions as taking only positive values and being skewed right, less so as the degrees of freedom increase, and the chi-square statistic as a measure of the distance between observed and expected counts relative to the expected counts.
• Identify whether a chi-square test for homogeneity or a chi-square test for independence is appropriate, referencing the populations and variables in context.
• Identify the null and alternative hypotheses for each test in words, in context.
• Justify the randomization, 10% and expected counts conditions, checking that all expected counts are at least 5.

Covers CED Topic 3.15: expected counts $\frac{\text{row total} \times \text{column total}}{\text{table total}}$, the chi-square statistic $\chi^2=\sum \frac{(\text{Observed}-\text{Expected})^2}{\text{Expected}}$ with $(\text{rows}-1)(\text{columns}-1)$ degrees of freedom, the p-value, and a conclusion in context.

• Calculate the expected counts for a two-way table, assuming the null hypothesis is true.
• Calculate the chi-square statistic, its degrees of freedom and its p-value using Table C or technology.
• Interpret the p-value of a chi-square test in context, as a probability computed assuming the null hypothesis is true.
• Justify a conclusion about the population (independence) or populations (homogeneity) by comparing the p-value with $\alpha$, without accepting the null hypothesis.

Covers CED Topic 4.2: t-distributions and degrees of freedom, why $t$ replaces $z$ when $\sigma$ is unknown, the one-sample t-interval for a population mean and for a population mean difference from matched pairs, its conditions, the interval $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ with $n-1$ degrees of freedom, the standard error and the margin of error.

• Describe t-distributions as symmetric and bell-shaped with heavier tails than the standard normal, approaching it as the degrees of freedom increase, and explain that they are used when $\sigma$ is unknown and $s$ is used instead.
• Identify the one-sample t-interval for a population mean, or for a population mean difference by analyzing the differences in a matched pairs design, and define the parameter in context with the order of subtraction.
• Justify the randomization, 10% and sample data conditions: the population is approximately normal, or $n \ge 30$, or the sample (or the sample of differences) shows no strong skewness or outliers when $n<30$.
• Calculate $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ with $df=n-1$, including for the mean difference $\bar{x}_d$ of matched pairs.
• Calculate the standard error $\frac{s}{\sqrt{n}}$ and the margin of error $t^*\frac{s}{\sqrt{n}}$.

Covers CED Topic 4.3: interpreting a confidence interval and confidence level for a population mean or mean difference in context, using the interval to justify claims, and how the confidence level and sample size affect its width.

• Interpret a C% confidence interval for a population mean or population mean difference in context, referring to the parameter.
• Interpret the confidence level in terms of repeated random sampling with the same sample size from the same population.
• Justify a claim about a population mean or mean difference based on the interval of plausible values.
• Identify how the confidence level and the sample size affect the critical value, the margin of error and the width of the interval.

Covers CED Topics 4.4–4.5: setting up and carrying out a one-sample t-test for a population mean, or for a population mean difference from matched pairs: parameter, hypotheses, conditions, the test statistic $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$ with $n-1$ degrees of freedom, the p-value and a conclusion in context.

• Identify the one-sample t-test for a population mean, or for a population mean difference in a matched pairs design, and define the parameter in context.
• Identify the null hypothesis $H_0: \mu=\mu_0$ (or $\mu_d=0$) and a one-sided or two-sided alternative.
• Justify the randomization, 10% and sample data conditions for the test.
• Calculate $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$ with $df=n-1$ and its p-value using Table B or technology, and interpret the p-value assuming $H_0$ is true.
• Justify a conclusion about the population in context by comparing the p-value with $\alpha$, in terms of the alternative and with nondefinitive language.

Covers CED Topic 4.6: the mean $\mu_1-\mu_2$ and standard deviation $\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$ of $\bar{x}_1-\bar{x}_2$, the conditions for independent random samples or randomized experiments, normality from normal populations or $n_1 \ge 30$ and $n_2 \ge 30$, and probabilities interpreted for two populations.

• Calculate the mean and standard deviation of the sampling distribution of $\bar{x}_1-\bar{x}_2$.
• Justify the randomization condition (two independent random samples, or random assignment in an experiment) and the 10% condition for each sample.
• Justify a normal or approximately normal model for $\bar{x}_1-\bar{x}_2$: both populations normal, or both sample sizes at least 30.
• Calculate and interpret probabilities involving $\bar{x}_1-\bar{x}_2$ in the context of the two populations.

Covers CED Topics 4.7–4.8: the two-sample t-interval for $\mu_1-\mu_2$: parameters, conditions, $(\bar{x}_1-\bar{x}_2) \pm t^*\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$ with degrees of freedom from technology, the standard error and margin of error, interpretation in context, and whether the interval contains 0.

• Identify the two-sample t-interval for a difference between population means and define the parameter in context, including the order of subtraction.
• Justify the randomization, 10% and sample data conditions for both samples.
• Calculate $(\bar{x}_1-\bar{x}_2) \pm t^*\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$, its standard error and its margin of error, with degrees of freedom from technology (between the smaller of $n_1-1$ and $n_2-1$ and $n_1+n_2-2$).
• Interpret the interval and the confidence level in context for the difference between two population means.
• Justify a claim about a difference in population means, using whether the interval contains 0 or another claimed value.

Covers CED Topics 4.9–4.10: the two-sample t-test for $\mu_1-\mu_2$: parameters, hypotheses, conditions, the test statistic $t=\frac{(\bar{x}_1-\bar{x}_2)-0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$ with degrees of freedom from technology, the p-value and a conclusion in context.

• Identify the two-sample t-test for a difference between population means and define the parameters in context.
• Identify the null hypothesis $H_0: \mu_1=\mu_2$ (or $\mu_1-\mu_2=0$) and a one-sided or two-sided alternative.
• Justify the randomization, 10% and sample data conditions for both samples.
• Calculate the two-sample t-statistic and its p-value using technology, and interpret the p-value assuming the population means are equal.
• Justify a conclusion about the populations in context by comparing the p-value with $\alpha$, in terms of the alternative and with nondefinitive language.

Technique for the 10-point inference question, a hypothesis test or a confidence interval using any procedure from Units 3 and 4: identifying and classifying variables, naming the procedure, stating parameters and hypotheses, verifying conditions in context, calculating results with work shown, and justifying a conclusion. Each component is a separately scored point.

• Determine the appropriate inference procedure by name (one- or two-sample z for proportions, chi-square for homogeneity or independence, one-sample, matched pairs or two-sample t for means) from the type of variable, the number of samples or groups and the goal (test or interval).
• Identify and classify the variables, and define the parameter or parameters and the population or populations in context.
• Verify each condition (randomization, the 10% condition when relevant, and normality, sample data or expected counts) explicitly with the numbers from the problem.
• Calculate the test statistic and p-value, or the interval, showing the structure of the formula and the substituted values in clear keyboard notation.
• Justify a conclusion that links the p-value to $\alpha$, or the interval to the claim, and answers the investigative question in context.

Covers CED Topic 5.2: interpreting the correlation coefficient $r$ as a unit-free measure of the strength and direction of a linear association, its limits, and why correlation does not imply causation.

• Interpret the sign and size of the correlation coefficient $r$, which is unit-free and between $-1$ and $1$, for a linear association in context.
• Explain that $r=0$ indicates no linear association and $r=-1$ or $r=1$ a perfect linear association.
• Explain why a correlation close to $-1$ or $1$ does not by itself show that a linear model is appropriate.
• Explain why an association, even a strong one, does not imply that changes in one variable cause changes in the other.

Covers CED Topic 5.3: using a linear regression model $\hat{y}=a+bx$ to calculate predicted responses, and distinguishing interpolation from extrapolation.

• Calculate a predicted response $\hat{y}=a+bx$ from a linear regression model for a given value of the explanatory variable.
• Explain when a linear regression model is a reasonable way to approximate the relationship, based on its form.
• Distinguish interpolation from extrapolation, and explain why predictions become less reliable the further they are extrapolated.

Covers CED Topic 5.4: calculating and interpreting residuals ($y-\hat{y}$), and using residual plots to judge whether a linear model is appropriate.

• Calculate a residual as observed $y$ minus predicted $\hat{y}$, and find an observed value from a residual and a prediction.
• Interpret a positive residual as the model underpredicting and a negative residual as the model overpredicting the response, in context.
• Describe a residual plot as a scatterplot of residuals against predicted values or against values of the explanatory variable.
• Describe the form of an association from a residual plot: apparent randomness supports a linear model, and curvature suggests a linear model is not the most appropriate.

Covers CED Topic 5.5: the least-squares regression line, found with technology by minimizing the sum of squared residuals, its slope $b$, y-intercept $a$, correlation $r$ and coefficient of determination $r^2$, and interpreting the coefficients in context.

• Calculate the slope $b$, y-intercept $a$, correlation $r$ and $r^2$ of the least-squares regression line using technology or given computer output.
• Explain that the least-squares line minimizes the sum of the squared residuals and passes through $(\bar{x}, \bar{y})$.
• Interpret the slope as the predicted increase or decrease in the response for a one-unit increase in the explanatory variable, in context.
• Interpret the y-intercept in context, and explain when it has no reasonable meaning because $x=0$ is an extrapolation or the predicted value is impossible.
• Interpret $r^2$ as the proportion of the variation in the response variable explained by the linear relationship with the explanatory variable.

Technique for the 10-point multi-part question that combines several content areas (for example probability, summary statistics and an inference procedure) and assesses Practices 2, 3 and 4: reading the whole scenario, linking parts, carrying results forward, and justifying claims with or without inference as the prompt requires.

• Identify which content area and practice each part addresses, from descriptive statistics and probability to inference.
• Calculate probabilities, means or other statistics from tables or distributions, showing the expression, the substitution and the answer.
• Distinguish a justification based on a descriptive comparison from one based on an inference procedure, and give the one the prompt asks for.
• Carry forward results from earlier parts when later parts build on them.
• Plan time across the four questions so that this final multi-part question receives full attention.